6.10:
Normal Distribution
The normal distribution is a continuous probability distribution with a symmetrical, bell-shaped graph. It is described by the Gaussian distribution formula with mean and standard deviation as the fixed parameters. The π and e are constant values.
Consider the birth weight of babies with a mean of 3.5 kg and a standard deviation of 0.4 kg. The data can be visualized by plotting probability density versus birth weight.
From the z score formula, birth weights can be standardized into corresponding z scores.
Re-plotting the probability density with z score shows that the graph is now centered around zero.
This standardized form of the normal distribution is known as the standard normal distribution, in which the mean is zero, and the standard deviation is one.
Such conversion of the normal distribution into standard normal distribution simplifies the Gaussian distribution formula, easing the calculation of probability values.
It is also useful for comparing data sets having different means and standard deviations.
6.10:
Normal Distribution
The normal, a continuous distribution, is the most important of all the distributions. Its graph is a bell-shaped symmetrical curve, which is observed in almost all disciplines. Some of these include psychology, business, economics, the sciences, nursing, and, of course, mathematics. Some instructors may use the normal distribution to help determine students’ grades. Most IQ scores are normally distributed. Often real-estate prices fit a normal distribution. The normal distribution is extremely important, but it cannot be applied to everything in the real world. The following equation describes this distribution:
Where μ represents the mean, σ is the standard deviation. The values of π and e are constant. The f(x) represents the probability of a random variable x.
The curve is symmetric about a vertical line drawn through the mean, μ. In theory, the mean is the same as the median, because the graph is symmetric about μ. As the notation indicates, the normal distribution depends only on the mean and the standard deviation. Since the area under the curve must equal one, a change in the standard deviation, σ, causes a change in the shape of the curve; the curve becomes fatter or skinnier depending on σ. A change in μ causes the graph to shift to the left or right. This means there are an infinite number of normal probability distributions. One of special interest is called the standard normal distribution.
The standard normal distribution is a normal distribution of standardized values called z scores. A z score is measured in units of the standard deviation. For example, if the mean of a normal distribution is five and the standard deviation is two, the value 11 is three standard deviations above (or to the right of) the mean.
This text is adapted from Openstax, Introductory Statistics, Section 6 Introduction.